05-67 Yuri Kozitsky

Irreducible Dynamics of Quantum Systems Associated with L{\'e}vy Processes (39K, AMS-LaTeX) Feb 14, 05

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Abstract. Quantum systems described by the Schr\"odinger operators $H = \sum_{j=1}^N \mathit{\Phi}(p_j) + W(x_1 , \dots, x_N)$, $p_j = -\imath \mathit{\nabla}_j$, $x_j \in \mathbb{R}^\nu$ with $\mathit{\Phi}$ being continuous functions such that the pseudo-differential operators $\mathit{\Phi}(p_j)$ generate L{\'e}vy processes, are considered. It is proven that the linear span of the operators $\alpha_{t_1}(F_1) \cdots \alpha_{t_n} (F_n)$ is dense in the algebra of all observables in the $\sigma$-strong and hence in the $\sigma$-weak and strong topologies. Here $\alpha_t (F) = \exp(\imath tH) F \exp(- \imath tH)$ are time automorphisms and $F$'s are taken from families of multiplication operators obeying conditions described in the paper. This result implies that a linear functional continuous in either of these topologies is fully determined by its values on such products. In the case of KMS states this yields a representation of such states in terms of path integrals.

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